Kuwata
Chemistry 222
Spring 2011
Analytical Chemistry
Constructing a Calibration Curve by the Method of Least Squares
A. First Iteration: Using Add Trendline
After you create the above spreadsheet, select the data in Columns A and C and generate a plot.
Next, click on the points, and do the following:
 Select “Add Trendline” under the Chart pull-down menu.
 Under the “Type” tab, choose a linear Trend/Regression Type.
 Under the “Options” tab, choose to display both the equation and R-squared (R2) value on
the chart.
 Click on your trendline box and go to “Selected Data Labels” in the Format pull-down
menu. Under the Number tab, choose to display at least three figures for your
parameters.
Calibration Curve
0.4000
y = 0.01630x + 0.00467
Signal
0.3000
2
R = 0.99785
0.2000
0.1000
0.0000
0
5
10
[Protein Standards] (ug)
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Chemistry 222
The correlation coefficient R2 is a good qualitative measure of linearity, but…
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Kuwata
Chemistry 222
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B. Second Iteration: Using the Excel Array Function LINEST
LINEST is an example of an array function with four arguments. In the above spreadsheet, you
would enter it as follows:
 Select a 2-column by 5-row array of cells (D20:E24 above) (Note the use of a colon to
specify a range of cells.)
 Type in =linest(c3:c16,a3:a16,true,true)
LINEST’s first argument is the range of cells containing y-values. The second argument
is the range of cells containing x-values. (Excel will complain if the number of y-values
does not match the number of x-values.) The third argument (true or false) refers to
whether we want to optimize the y-intercept (true) or force the y-intercept to be zero
(false). The fourth argument (true or false) is asking if we want other statistical
parameters besides m and b. Always say true for the last two arguments.


(On Windows machines:) Press CTRL-SHIFT-ENTER simultaneously
(On Macintoshes:) Press OpenApple-SHIFT-ENTER simultaneously
The above spreadsheet labels seven of the ten parameters computed by LINEST. It reports not
only the least squares parameters m and b, but also the standard errors of measurement in m (that
is, sm), in b (that is, sb), and in a reading y made on a sample (that is, sy). Because these are
standard errors of measurement (that is, standard deviations divided by n ), you obtain 95%
confidence intervals for m, b, and y simply by multiplying sm, sb, and sy by the appropriate value
of Student’s t for n-2 degrees of freedom. We lose two degrees of freedom since we have
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Kuwata
Chemistry 222
Spring 2011
calculated both a slope and a y-intercept from the data. (Note that Harris is wrong: LINEST does
not report standard deviations in m, b, and y: they have already been divided by n .)
The standard error in the slope is enough information in many cases (such as in Physical
Chemistry I experiments), but in Analytical Chemistry, we want to quantify the error in x, the
concentration corresponding to a measurement y….
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Chemistry 222
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Spring 2011
Kuwata
Chemistry 222
Spring 2011
C. Final Iteration: Treating the Correlation in the Errors in Slope and Y-Intercept
(also see spreadsheet in Harris Figure 4-13)
For k measurements on an unknown, we get an average signal y. We solve for the unknown’s
concentration x. We then calculate the standard error of measurement in x thus:
sx 
sy
m
1 1
( y  y) 2
 
k n m 2  ( xi  x ) 2
As before, you compute 95% confidence intervals by multiplying sx by the appropriate value of
Student’s t for n-2 degrees of freedom. (Note that while a larger value of k increases the
precision of our determination of x, it does not affect how many degrees of freedom we have.)
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Chemistry 222
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Spring 2011
Kuwata
Chemistry 222
Spring 2011
The shaded area shows the standard errors in x (sx) computed correctly, that is, by treating the
correlation in sm and sb. Note how the errors increase as one gets further away from the
calibration curve’s centroid.
(Taken from the 6th edition of Harris’ Quantitative Chemical Analysis.)
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